Vucic has said his change of job would not alter the former Yugoslav republic's geopolitical balance between the European Union, which Serbia wants to join, and Russia.
Hundreds of people protested in Belgrade on Tuesday against the overwhelming victory of Prime Minister Aleksandar Vucic in a presidential election.
Vucic took a decisive 55.02 percent of votes in the election, confirming his domination over the Balkan country as he pursues a balancing act between Europe and Russia.
His new job will be largely ceremonial, but he is expected to retain de facto power through control of his ruling Serbian Progressive Party.
People rallied in front of the Serbian parliament, following announcements on social media, chanting "No to dictatorship."
"What we want is Vucic to step down, he is an autocrat, and we do not want him," said Nemanja Tomic, 22, a student. Some demonstrators alleged Sunday's presidential vote was rigged.
The election result came as a humiliation for the beleaguered opposition.
Sasa Jankovic, the main opposition candidate and former rights advocate, came second with only 16.36 percent of votes.
Twenty-five-year-old student Luka Maksimovic, who ran as a white-suited parody of a sleazy imaginary politician called Ljubisa "Beli" Preletacevic, came third with 9.43 percent.
Both Jankovic and Maksimovic have expressed support for the protests but have not joined them.